Extrasolar planets are difficult to detect due to their vast distance from Earth, but a few methods are available to find them.

Doppler SpectroscopyStarlight is sorted by wavelength into a spectrum ("rainbow"). Atoms and molecules in the star absorb certain wavelengths, leaving dark lines in the spectrum called "absorption lines." If the star is host to an orbiting planet, the planet's gravitational pull on the star will cause it to orbit around the system's centre of mass. The motion of the star makes itself apparent by imparting a Doppler shift in its spectrum, resulting in an apparent change in the position of the absorption lines. This permits you to detect the stellar radial velocity - the velocity of the star along the line of sight. This will have a sinusoidal variation that can be easily modelled in terms of Keplerian motion of the star due to an orbiting body, yielding the planet's minimum separation, orbital period, eccentricity, and minimum mass.

This method is responsible for the discovery of the majority of planets detected to date, and requires precise spectrometers. Examples include Keck's HIRES spectrometer and ESO's HARPS spectrometer, both of which are very productive exoplanet discovery engines.

Transit MethodThis is currently the most common method of discovering extrasolar planets (and took the lead over Doppler spectroscopy last year). A (small) fraction of exoplanets will have orbits aligned in such a way to where they will transit in front of their stars as seen from Earth, resulting in a transient decrease in the observed system brightness.

If a planet is detected by both transits and radial velocity, then the planet's true mass is known, and the radius can be deducted by the transit depth, yielding the planetary density and therefore a rough guess of its bulk composition.

Because transits are short-duration events, lasting a few hours and occuring once per planetary orbital period, it is helpful to have continuous coverage of a star to confidently detect any transiting planets it has (as opposed to getting "lucky" and observing the right star at the right time). While ground-based surveys like TrES, XO, Super-WASP and HATNet, space-based observatories like Kepler and CoRoT are able to monitor stars continuously, making them more sensitive to planets in longer orbital periods.

MicrolensingWith the path of light being purturbed by gravity, gravitational fields can act as giant "lenses," magnifying a background light source. Because stars are in motion, these "microlensing" events are transient. The background star follows a characteristic brightening before dimming.

If the foreground lensing star is a multi-component system - perhaps a binary star or a star with a planetary companion), the characteristic lensing profile can be distorted in a way that can be modelled.

Groups using this method include OGLE and MOA, which have telescopes spread around the world to provide continuous coverage of microlensing events.

AstrometryAstrometry is very straightforward, but in practice very hard to do at the precision level needed to detect planets. The motion of a star orbited by planets around the system's centre of mass is an observable phenomenon, and can be detected directly to determine the full, 3D orbit of the planet.

The upcoming GAIA spacecraft will use astrometry to determine the parallaxes, motions and colours of a billion stars, and is expected to find thousands of giant planets in < 10 yr orbits within 200 pc.

Direct ImagingIn some cases, it is possible to directly detect the planet with imaging. The difficulties facing this method are immense - planets are much dimmer than, and right next to, their host stars, placing demanding contrast requirements on the instrumentation. This has been performed in the infrared (where the contrast ratio is lower and the planet brightness more easily scales with mass) on only a few planets with high separations from their host stars and high masses.

The two major detection methods - radial velocity and transit photometry - are prone to biases.

From a purely geometric perspective, the probability that a planet will transit is determined by how far away it is from the star, as well as the size of the star. This relationship is written

Where R_* is the stellar radius and a is the separation between the two bodies. We can see that as a increases, the probability that a planet will transit decreases, and therefore the overwhelming majority of planets which transit will be in short-period orbits.

The sensitivity of a planet to detection by radial velocity can be thought of in terms of the amplitude of the velocity of the star throughout its barycentric motion around the system's centre of mass, expressed as

Where a_* is the semi-major axis of the stellar orbit (itself a function of the mass and separation of the planet via Kepler's Laws), i is the inclination of the orbit with respect to the plane of the sky, such that i=90 degrees represents an edge-on orbit, P is the orbital period of the system, and e is the eccentricity of the orbits. The period and eccentricity of the stellar orbit will be identical to the period and eccentricity of the planetary orbit.

In the case of Doppler spectroscopy, since the detectability of the planet is not contingent on the inclination being in a small range of values, the method is much more sensitive to planets in longer periods. It is still biased toward short-period planets, but to less of a degree.

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